Talking About Convergence
For most of my college career I was a hard-core syntax wienie,Ã¢â‚¬ David Foster Wallace confessed to an interviewer in 1993, describing his transformation from a talented analytic philosophy major to the literary wunderkind who would garner comparisons to Joyce and Pynchon at age 25 with his first novel, The Broom of the System. His switch from dense logic puzzles to hyperactive prose came when, burnedout with counterfactuals, he took a semester off from Amherst, read a ton of literature, and started dabbling in fiction. What he discovered was that the buzz heÃ¢â‚¬â„¢d gotten when a proof finally came togetherÃ¢â‚¬”the moment when he suddenly saw Ã¢â‚¬Å“a gorgeously simple solutionÃ¢â‚¬Â¦ after filling half a notebook with gnarly attemptÃ¢â‚¬Ã¢â‚¬”was aesthetic in nature, and available in prose too. Concise, well-made sentences, his own and othersÃ¢â‚¬â„¢, produced in him the same unmistakable sensation elegant math maneuvers had. Wallace called the feeling Ã¢â‚¬Å“the click,Ã¢â‚¬ and he hears it, he told the interviewer, in Hopkins and Donne, in most Nabokov, Ã¢â‚¬Å“almost line by lineÃ¢â‚¬ in Delillo. Ã¢â‚¬Å“Puig clicks like a fucking Geiger counter,Ã¢â‚¬ he noted, adding that the click is Ã¢â‚¬Å“maybe the only way to describe writers I love.Ã¢â‚¬
Two years after that interview he would publish his magnum opus Infinite Jest, 1,079 pages of harebrained clicking that filters HamletÃ¢â‚¬â„¢s meditation on the skull of Yorick (Ã¢â‚¬Å“a fellow of infinite jestÃ¢â‚¬) through wicked layers of illicit pharmacology, deadly entertainment, and super-addictive personalities. It remains for some brilliant grad student of the future to parse out all the connections between that title and Everything and More, a popular history of infinityÃ¢â‚¬â„¢s technical meaning written for NortonÃ¢â‚¬â„¢s brand-new Great Discoveries series (and the higher-math dissertation Wallace the novelist never got to write). But for now even just a casual look through this latest book will probably give readers of WallaceÃ¢â‚¬â„¢s fiction and essays a sense of deja vu: pages with three lines of main text at the top and four long footnotes (some marked IYI, for Ã¢â‚¬Å“If youÃ¢â‚¬â„¢re interestedÃ¢â‚¬) rising up from the bottom; bold-faced headings on the order of Ã¢â‚¬Å“Emergency Glossary,Ã¢â‚¬ Ã¢â‚¬Å“Mini-Interpolation,Ã¢â‚¬ and Ã¢â‚¬Å“Quick Forest-V.-Tree InterpolationÃ¢â‚¬; and flip phrasings in abundance like Ã¢â‚¬Å“So the thing is now weÃ¢â‚¬â„¢re talking about convergence.Ã¢â‚¬
All this comes wrapped in an argument that, necessarily, circles and circles, stating, refining, re-stating, asking you to skip or review certain sections as your math knowledge warrants, holding off on this for now, diving into that (though we probably donÃ¢â‚¬â„¢t need it at the momentÃ¢â‚¬”but if youÃ¢â‚¬â„¢re interested…). As I took WallaceÃ¢â‚¬â„¢s usual stylistic challenge of holding open multiple story-lines and logical sequences at once (a challenge deepened considerably by the abstractness of difficult math), I began to think: Maybe Wallace the syntax wienie has been with us all along, writing weird, beautiful proofs and calling them novels, stories, and profiles of David Lynch.
The star of this latest dazzling narrative is (aside from the author himself) Georg Cantor (1845-1918), German mathematician, sometime mental patient, and creator of set theory and transfinite numbers (and, by extension, Wallace says, most all of interest in 20th-century math and logic). Readers should be warned not to expect a biography full of wry anecdotes. While pausing occasionally for quick sketches of infighting cranks and their unwieldy 32-word paper titles, Wallace keeps the focus on defining key terms and explicating proofs, critical of the tendency of other pop histories to belabor CantorÃ¢â‚¬â„¢s mental health and reductively assume it was infinity that put him in an asylum for the last several years of his life. In WallaceÃ¢â‚¬â„¢s persuasive account, CantorÃ¢â‚¬â„¢s illness (probably manic-depression, and most acute after his major work was complete) obscures the fact that he actually put this most mind-bending of concepts on a rational footing. Thus Wallace stops only briefly, in a late footnote, over CantorÃ¢â‚¬â„¢s sad state at his death: oblivious to the once-dismissive math community now crowning him a genius, obsessed not by infinity but by JesusÃ¢â‚¬â„¢s biological paternity and the Shakespeare v. Bacon controversy.
In fact, Cantor himself figures only as a horizon point in WallaceÃ¢â‚¬â„¢s larger historical task: to convince us that infinity haunted math for 2,000+ years as an absolutely necessary concept which a Greek-inspired prejudice for Ã¢â‚¬Å“rationalÃ¢â‚¬ numbers kept pushing underground. As it tells this story in its first 200 pages, I found, the book operates on a kind of gradual slope that turnsÃ¢â‚¬”inevitably, for the last hundredÃ¢â‚¬”very, very steep. ThatÃ¢â‚¬â„¢s when Cantor and his contemporaries enter and, via the critical idea of the set, definitively prove the existence of not just infinity but multiple infinitiesÃ¢â‚¬”powers of infinities, one larger than the next, quantities of a different order from those weÃ¢â‚¬â„¢re used to and yet capable of being (under special circumstances I donÃ¢â‚¬â„¢t understand at all) added and subtracted.
Ã¢â‚¬Å“Existence,Ã¢â‚¬ did you say? Infinity was discovered? Like one discovers an actual object? Yes, Cantor and Wallace resoundingly answer. The iconoclastic Cantor was an unforgiving mathematical Platonist, meaning he regarded quantities and their relations as extra-mentally real, that is, not merely created by the human mind for its ordering purposes. Infinity is (in WallaceÃ¢â‚¬â„¢s recurrent example) more horse than made-up unicorn. The spokesman for the opposing camp (and thus the big villain in WallaceÃ¢â‚¬â„¢s story) is Aristotle, who in his Metaphysics, along with refuting PlatoÃ¢â‚¬â„¢s theory of forms, insisted that infinity could only ever be Ã¢â‚¬Å“potential,Ã¢â‚¬ never Ã¢â‚¬Å“actualÃ¢â‚¬Ã¢â‚¬”that is, if we canÃ¢â‚¬â„¢t ever possibly count it out, it canÃ¢â‚¬â„¢t be a number and canÃ¢â‚¬â„¢t, per se, exist. AristotleÃ¢â‚¬â„¢s target on the infinity question was the infamous Zeno of ZenoÃ¢â‚¬â„¢s Paradoxes, which Wallace summarizes this way: To cross the street, you obviously first need to get halfway across. And before that, halfway to that halfway point, and before that, halfway to halfway to halfway. In other words, any two points I can specify on a line will always have one between them, and you never get across the street. Thus ZenoÃ¢â‚¬â„¢s Vicious Infinite Regress, which Wallace abbreviates (with his obvious glee for abbreviations) V.I.R.
AristotleÃ¢â‚¬â„¢s grip on infinity grows stronger over the next 1500 years as the Church, helped along by Aquinas, claims that only God is actually infinite and nothing else in His creation can be. But Zeno and his V.I.R. hang around, proving to be what Wallace persistently calls the major Ã¢â‚¬Å“crevasseÃ¢â‚¬ in math logic. Until, in the 1660s (to cut a fascinating tale quite short), Newton and Leibniz nearly simultaneously invent the calculus. And, as legions of fearful math students were thereby fated to learn every September, the basic operation of the calculus is to divide the area under a curve into infinitely many pieces. But divide by infinity without first acknowledging or proving it exists? In that disjunction begins a massive, centuries-long agon within math circles, pitting the incredibly useful results of the calculus (for physics, astronomy, engineeringÃ¢â‚¬”everything) against the pressing need (this being math, after all) to give those new methods a firm logical foundation in proofs. The latter camp will leadÃ¢â‚¬”eventually, after much, much intermediate hair-splittingÃ¢â‚¬”to Cantor. The former camp, meanwhile, keeps accepting what works and challenging the foundationalists to keep pace, operating on a principle voiced by a particularly practical mathematician when heÃ¢â‚¬â„¢s asked to rigorously prove the concept of a limit: Ã¢â‚¬Å“Just keep moving forward and faith will come to you.Ã¢â‚¬
To sound a note Wallace does countless times himself: ItÃ¢â‚¬â„¢s actually way, way more complicated than that. Not to mention that the above only sets the stage for the stunning advances of set theory that are the bookÃ¢â‚¬â„¢s true destinationÃ¢â‚¬”and reduces to quick and crude natural language the distinctions that Wallace spends pages (and the mathematicians spent years) trying to Ã¢â‚¬Å“arithmetize,Ã¢â‚¬ or put in the fool-proof, universal terms of math. Fair warning, then: This book is extremely heavy going in stretches, even if you find Wallace, tics and all, to be the most lucid and conscientious of explainers writing these days, as I do. He has a logicianÃ¢â‚¬â„¢s uncanny sense for where his language drops even slightly into opacity, andÃ¢â‚¬”on rap, television, and discontinuous functionsÃ¢â‚¬”I would rather be in his hands than in anyone elseÃ¢â‚¬â„¢s.
Still, I admit to doing more smiling and nodding than understanding on the finer points, particularly in the heavy-duty final third. About 250 pages in, Wallace echoes his definition of the click in describing a particularly knotty step Cantor takes: Ã¢â‚¬Å“The proof is both ingenious and beautifulÃ¢â‚¬”a total confirmation of artÃ¢â‚¬â„¢s compresence in pure math.Ã¢â‚¬ Beauty, I wearily agreed at that moment, is definitely in the eye of the beholder. Or to put it another way, the click sounds most clearly in the ear of he who has more than a 12-years-gone memory of AP calculus with which to hear it. But with that caveat, this is a remarkable book, and worth putting even a neophyteÃ¢â‚¬â„¢s ear to.
Jeff Severs is a writer in Austin.